12/03/2019
Common to pool data or consider problems on a region by region basis.
This can make statistical problems more tractable.
National Resource Management Regions
CSIRO and Bureau of Meteorology, 2015. Climate change in Australia information for Australia's natural resource management regions: Technical report.
Whan, Kirien, and Maurice Schmeits. "Comparing area probability forecasts of (extreme) local precipitation using parametric and machine learning statistical postprocessing methods." Monthly Weather Review 146.11 (2018): 3651-3673.
How should assign regions for the analysis of extremes?
Create regions that are likely to experience similar impacts
These regions can then inform our statistical analysis
1. Regionalisation
2. Visualise spatial dependence
3. Spatial post-processing
Require: Notion of closeness between two locations
Want: Form clusters based on extremal dependence
Solution: The F-madogram distance
Bernard, Elsa, et al. "Clustering of maxima: Spatial dependencies among heavy rainfall in France." Journal of Climate 26.20 (2013): 7929-7937.
\[d(x_i, x_j) = \tfrac{1}{2} \mathbb{E} \left[ \left| F_i(M_{x_i}) - F_j(M_{x_j})) \right| \right]\] where \(M_{x_i}\) is the annual maximum rainfall at location \(x_i \in \mathbb{R}^2\) and \(F_i\) is the distribution function of \(M_{x_i}\).
Advantages:
Cooley, D., Naveau, P. and Poncet, P., 2006. Variograms for spatial max-stable random fields. In Dependence in probability and statistics (pp. 373-390). Springer, New York, NY.
For \(M_{x_i}\) and \(M_{x_j}\) with common GEV marginals, \(\theta(x_i - x_j)\) is \[\mathbb{P}\left( M_{x_i} \leq z, M_{x_j} \leq z \right) = \left[\mathbb{P}(M_{x_i}\leq z)\mathbb{P}(M_{x_i}\leq z)) \right]^{\tfrac{1}{2}\theta(x_i - x_j)}. %= \exp\left(\dfrac{-\theta(h)}{z}\right),\]
The range of \(\theta(x_i - x_j)\) is \([1 , 2]\).
Can write our distance measure as a function of the extremal coefficient, \(\theta(x_i - x_j)\), \[d(x_i, x_j) = \dfrac{\theta(x_i - x_j) - 1}{2(\theta(x_i - x_j) + 1)}.\]
Therefore the range of \(d(x_i, x_j)\) is \([0 , 1/6]\).
Kaufman, L. and Rousseeuw, P.J., 1990. Partitioning around medoids (PAM). Finding groups in data: an introduction to cluster analysis, pp.68-125.
Consider the \(\max \{ \| x_i - x_j \|, 2\}\) as the clustering distance.
Spatial density is changed by land-sea and domain boundaries
Tendancy toward clusters of equal size
Clustering is in F-madogram space not Euclidean
SHINY APP
IMAGE
Where can we assume a common dependence structure for extremes?
Shiny App
SWWA
TAS
Oesting et. al 2017
approach
cut the region into two